Suppose a 69 kg person stands at the edge of a 7.8 m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 1600 kg*m^2. The turntable is at rest initially, but when the person begins running at a speed of 3.7 m/s (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable. Please help with step-by-step explanation.

The total angular momentum remains zero because of the frictionless bearings. The person and the turntable rotate about the axis of rotation in opposite directions.

Let w be the angular velocity of the turntable after "the person" begins running. His (or her) angular momentum about the axis is
M V R = M (3.7 - R w) *R
Note that we have to use the speed V with respect to land, not the turntable. That is why R w has to be subtracted from the velocity with respect to the turntable.